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Theorem ltexpri 7673

Description: Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)

**Assertion**
Ref	Expression
ltexpri	⊢ (𝐴<_P 𝐵 → ∃𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

Proof of Theorem ltexpri Dummy variables 𝑦 𝑧 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . . . 8 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
2	1	eleq1d 2262	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝑧 ∈ (2^nd ‘𝐴) ↔ 𝑣 ∈ (2^nd ‘𝐴)))
3		simpl 109	. . . . . . . . 9 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → 𝑦 = 𝑢)
4	1, 3	oveq12d 5936	. . . . . . . 8 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝑧 +_Q 𝑦) = (𝑣 +_Q 𝑢))
5	4	eleq1d 2262	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵) ↔ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵)))
6	2, 5	anbi12d 473	. . . . . 6 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵)) ↔ (𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))))
7	6	cbvexdva 1941	. . . . 5 ⊢ (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵)) ↔ ∃𝑣(𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))))
8	7	cbvrabv 2759	. . . 4 ⊢ {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))} = {𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))}
9	1	eleq1d 2262	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝑧 ∈ (1^st ‘𝐴) ↔ 𝑣 ∈ (1^st ‘𝐴)))
10	4	eleq1d 2262	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵) ↔ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵)))
11	9, 10	anbi12d 473	. . . . . 6 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵)) ↔ (𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))))
12	11	cbvexdva 1941	. . . . 5 ⊢ (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵)) ↔ ∃𝑣(𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))))
13	12	cbvrabv 2759	. . . 4 ⊢ {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))} = {𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))}
14	8, 13	opeq12i 3809	. . 3 ⊢ ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ = ⟨{𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))}, {𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))}⟩
15	14	ltexprlempr 7668	. 2 ⊢ (𝐴<_P 𝐵 → ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ ∈ P)
16	14	ltexprlemfl 7669	. . . 4 ⊢ (𝐴<_P 𝐵 → (1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) ⊆ (1^st ‘𝐵))
17	14	ltexprlemrl 7670	. . . 4 ⊢ (𝐴<_P 𝐵 → (1^st ‘𝐵) ⊆ (1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)))
18	16, 17	eqssd 3196	. . 3 ⊢ (𝐴<_P 𝐵 → (1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (1^st ‘𝐵))
19	14	ltexprlemfu 7671	. . . 4 ⊢ (𝐴<_P 𝐵 → (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) ⊆ (2^nd ‘𝐵))
20	14	ltexprlemru 7672	. . . 4 ⊢ (𝐴<_P 𝐵 → (2^nd ‘𝐵) ⊆ (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)))
21	19, 20	eqssd 3196	. . 3 ⊢ (𝐴<_P 𝐵 → (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (2^nd ‘𝐵))
22		ltrelpr 7565	. . . . . . 7 ⊢ <_P ⊆ (P × P)
23	22	brel 4711	. . . . . 6 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
24	23	simpld 112	. . . . 5 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
25		addclpr 7597	. . . . 5 ⊢ ((𝐴 ∈ P ∧ ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ ∈ P) → (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) ∈ P)
26	24, 15, 25	syl2anc 411	. . . 4 ⊢ (𝐴<_P 𝐵 → (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) ∈ P)
27	23	simprd 114	. . . 4 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
28		preqlu 7532	. . . 4 ⊢ (((𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵 ↔ ((1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (1^st ‘𝐵) ∧ (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (2^nd ‘𝐵))))
29	26, 27, 28	syl2anc 411	. . 3 ⊢ (𝐴<_P 𝐵 → ((𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵 ↔ ((1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (1^st ‘𝐵) ∧ (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (2^nd ‘𝐵))))
30	18, 21, 29	mpbir2and 946	. 2 ⊢ (𝐴<_P 𝐵 → (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵)
31		oveq2 5926	. . . 4 ⊢ (𝑥 = ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ → (𝐴 +_P 𝑥) = (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩))
32	31	eqeq1d 2202	. . 3 ⊢ (𝑥 = ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ → ((𝐴 +_P 𝑥) = 𝐵 ↔ (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵))
33	32	rspcev 2864	. 2 ⊢ ((⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ ∈ P ∧ (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵) → ∃𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)
34	15, 30, 33	syl2anc 411	1 ⊢ (𝐴<_P 𝐵 → ∃𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∃wex 1503 ∈ wcel 2164 ∃wrex 2473 {crab 2476 ⟨cop 3621 class class class wbr 4029 ‘cfv 5254 (class class class)co 5918 1^st c1st 6191 2^nd c2nd 6192 Qcnq 7340 +_Q cplq 7342 Pcnp 7351 +_P cpp 7353 <_P cltp 7355

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-eprel 4320 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-1o 6469 df-2o 6470 df-oadd 6473 df-omul 6474 df-er 6587 df-ec 6589 df-qs 6593 df-ni 7364 df-pli 7365 df-mi 7366 df-lti 7367 df-plpq 7404 df-mpq 7405 df-enq 7407 df-nqqs 7408 df-plqqs 7409 df-mqqs 7410 df-1nqqs 7411 df-rq 7412 df-ltnqqs 7413 df-enq0 7484 df-nq0 7485 df-0nq0 7486 df-plq0 7487 df-mq0 7488 df-inp 7526 df-iplp 7528 df-iltp 7530

This theorem is referenced by: lteupri 7677 ltaprlem 7678 ltaprg 7679 ltmprr 7702 recexgt0sr 7833 mulgt0sr 7838 map2psrprg 7865

W3C validator